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Book review

Author(s) Retherford, J. R.
Title Hilbert Space: Compact Operators and the Trace Theorem
Publisher Cambridge University Press
Year of publication 1993
Reviewed by M. MARINOV

This nice book has been published as Volume 27 in the series of London Mathematical Society Student Texts, aimed at graduate (and advanced undergraduate) students of mathematics and theoretical physics. In my opinion, the book serves its purpose perfectly. The text is based upon an 8-week course given by the author, and its goal was to introduce some Hilbert (and Banach) space techniques. The required background includes mainly linear algebra and advanced calculus.

Methods of functional analysis in Hilbert space are widely employed in

applied mathematics and theoretical physics. The finite-dimensional linear algebra provides with usually helpful, but sometimes misleading, guidelines to the theory of linear operators. Intuitive statements need specifications and proofs. The conventional toolbox consists of a number of inequalities, bearing great names (Cauchy, Riesz, Hölder, Minkowski, Weyl, Hadamard, ...). The author shows them at work, explains the theories of bounded, self-adjoint, compact linear operators in the Hilbert space and the elements of the spectral theory. The Hilbert-Schmidt and trace operators are introduced and discussed. In conclusion, a simple proof is presented to the theorem by V. B. Lidskii (1959) on non-self-adjoint operators with a trace. The theorem is a natural extension of its prototypes in linear algebra and in the theory of self-adjoint operators. Namely, for a linear operator L in a separable Hilbert space its trace is defined as <formula>, for an orthonormal basis <formula>. It is proved that <fomula>, where <formula> are all eigenvalues of L.

The readers will find a number of exercises, hints and conjectures, making the text invaluable for an active study. The study would help a novice approaching fundamental treatises like those by Riesz and Nagy, Halmos, Dunford and Schwartz, König. Lecturers would also benefit greatly from this elegant and highly pedagogical introduction to operator methods.